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Corrected Kriging Update Formulae for Batch-Sequential Data Assimilation

  • Clément Chevalier
  • David Ginsbourger
  • Xavier Emery
Conference paper
Part of the Lecture Notes in Earth System Sciences book series (LNESS)

Abstract

Recently, a lot of effort has been spent in the efficient computation of kriging predictors when observations are assimilated sequentially. In particular, kriging update formulae enabling significant computational savings were derived. Taking advantage of the previous kriging mean and variance computations helps avoiding a costly matrix inversion when adding one observation to the \(n\) already available ones. In addition to traditional update formulae taking into account a single new observation, Emery (2009) proposed formulae for the batch-sequential case, i.e. when \(k\,>\,1\) new observations are simultaneously assimilated. However, the kriging variance and covariance formulae given in Emery (2009) for the batch-sequential case are not correct. In this paper, we fix this issue and establish correct expressions for updated kriging variances and covariances when assimilating observations in parallel. An application in sequential conditional simulation finally shows that coupling update and residual substitution approaches may enable significant speed-ups.

Keywords

Gaussian process  Kriging weights Sequential conditional simulation 

Notes

Acknowledgments

Part of this work has been conducted within the frame of the ReDice Consortium, gathering industrial (CEA, EDF, IFPEN, IRSN, Renault) and academic (Ecole des Mines de Saint-Etienne, INRIA, and the University of Bern) partners around advanced methods for Computer Experiments. Clément Chevalier also gratefully acknowledges support from the French Nuclear Safety Institute (IRSN) and warmly thanks Prof. Julien Bect for fruitful discussions on Gaussian process simulations. David acknowledges support from the IMSV.

References

  1. 1.
    Chevalier, C., & Ginsbourger, D. (2012): Corrected kriging update formulae for batch-sequential data assimilation. http://arxiv.org/abs/1203.6452
  2. 2.
    Chilès, J. P., & Delfiner, P. (2012). Geostatistics: Modeling spatial uncertainty (2nd edn.). New York: Wiley.Google Scholar
  3. 3.
    Emery, X. (2009). The kriging update equations and their application to the selection of neighboring data. Computational Geosciences, 13(1), 211–219.CrossRefGoogle Scholar
  4. 4.
    Hoshiya, M. (1995). Kriging and conditional simulation of gaussian field. Journal of Engineering Mechanics, 121(2), 181–186.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Clément Chevalier
    • 1
  • David Ginsbourger
    • 1
  • Xavier Emery
    • 2
  1. 1.IMSVUniversity of BernBernSwitzerland
  2. 2.Department of Mining Engineering and Advanced Mining Technology CenterUniversity of ChileSantiagoChile

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